Non-Transitory Computer Readable Recording Medium, Simulation Method and Simulation Device

ABSTRACT

A computer program causes a computer to execute processing of simulating behavior of an electromagnetic component including a coil at each of a plurality of time points based on an analytic model of the electromagnetic component. The processing comprises creating a look-up table storing a flux linkage of the coil, an inductance of the coil and a current in the coil that are obtained by a magnetic field analysis based on the analytic model in association with one another, and simulating behavior of the electromagnetic component by referring to the look-up table using currents in the coil calculated at a previous simulation step and at a step before the previous simulation step.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation application of PCT InternationalApplication No. PCT/JP2020/024259, which has an International filingdate of Jun. 19, 2020 and designated the United States of America, andclaiming priority on Patent Application No. 2020-082054 filed in Japanon May 7, 2020.

FIELD

The present invention relates to a computer program, a simulation methodand a simulation device that simulates the dynamic behavior of anelectromagnetic component including a coil based on an analytic model ofthe electromagnetic component.

BACKGROUND

For development of a motor and a driving circuit, a simulation devicefor simulating the dynamic behavior of a motor has been used. In orderto specifically and accurately simulate the behavior of a motor, a motorbehavior simulator that simulates the behavior of a motor using thecharacteristics obtained by a magnetic field analysis and a drivingcircuit simulator that simulates the operation of a driving circuit ofthe motor are configured to be coupled. In the coupled simulator, thedriving circuit simulator invokes the motor behavior simulator for eachsimulation step corresponding to each of the time points in a timeseries to specifically simulate the behavior of the motor, and simulatesthe behavior of the driving circuit by using the simulation results.

The motor behavior simulator previously creates and stores a look-uptable (LUT) showing the characteristics such as flux linkages, etc.depending on the driving state by performing an magnetic field analysison the analytic model representing the shape and electromagneticcharacteristics of the motor formed by multiple coils, a stator and arotor.

The motor behavior simulator simulates the behavior of the motor byreferring to the LUT obtained by the magnetic field analysis.

The motor is a star-connected three-phase permanent magnet motor, forexample. In this case, the motor behavior simulator solves voltageequations expressed by Equations (1)-(3) below to thereby calculatecurrents in the coils of the motor and simulates the behavior of themotor.

$\begin{matrix}{{V_{u\; 0} - V_{n}} = {{RI}_{u} + \frac{d\;{\Psi_{u}\left( {I_{u},I_{v},I_{w},\theta} \right)}}{dt}}} & (1) \\{{V_{v\; 0} - V_{n}} = {{RI}_{v} + \frac{d\;{\Psi_{v}\left( {I_{u},I_{v},I_{w},\theta} \right)}}{dt}}} & (2) \\{{V_{w\; 0} - V_{n}} = {{RI}_{w} + \frac{d\;{\Psi_{w}\left( {I_{u},I_{v},I_{w},\theta} \right)}}{dt}}} & (3)\end{matrix}$

where V_(uθ), V_(v0), V_(w0): voltage to be applied to the terminal ofeach coil

-   V_(n): voltage at the neutral point-   R: electrical resistance of each coil-   I_(u), I_(v), I_(w): current flowing in each coil-   Ψ_(u), Ψ_(v), Ψ_(w): flux linkage in each coil-   θ: mechanical angle of the rotor

Furthermore, the principle of current conservation shown in Equation (4)below holds at the neutral point.

I _(u) +I _(v) +I _(w)=0   (4)

The flux linkage Ψ is a function of a current and a mechanical angle ofthe rotor. Solving the nonlinear equations expressed by Equations(1)-(3) above generally needs repetitive calculation such as aNewton-Raphson method (see Hiroyuki Kaimori, Kan Akatsu, BehaviorModeling of Permanent Magnet Synchronous Motors Using Flux Linkages forCoupling with Circuit Simulation, IEEJ Journal of Industry Applications,Japan, The institute of Electrical Engineers of Japan, Vol. 7, No. 1 pp.56-63, for example).

Such repetitive calculation is a problem in the case of simulating anelectromagnetic component including a coil as well.

SUMMARY

In the conventional method, there was a problem of the need forrepetitive calculations to solve the voltage equations as describedabove.

An object of the present disclosure is to provide a computer program, asimulation method and a simulation device that are able to simulate thedynamic behavior of an electromagnetic component by solving nonlinearvoltage equations without performing repetitive calculation.

A computer program according to the present disclosure causes a computerto execute processing of simulating behavior of an electromagneticcomponent including a coil at each of a plurality of time points basedon an analytic model of the electromagnetic component. The processingcomprises: creating a look-up table storing a flux linkage of the coil,an inductance of the coil and a current in the coil that are obtained bya magnetic field analysis based on the analytic model in associationwith one another, and simulating behavior of the electromagneticcomponent by referring to the look-up table using currents in the coilcalculated at a previous simulation step and at a step before theprevious simulation step.

A simulation method according to the present disclosure is a simulationmethod causing a computer to execute processing of simulating behaviorof an electromagnetic component including a coil at each of a pluralityof time points based on an analytic model of the electromagneticcomponent, the processing comprising: creating a look-up table storing aflux linkage of the coil, an inductance of the coil and a current in thecoil that are obtained by a magnetic field analysis based on theanalytic model in association with one another; and simulating behaviorof the electromagnetic component by referring to the look-up table usingcurrents in the coil calculated at a previous simulation step and at astep before the previous simulation step.

A simulation device according to the present disclosure comprises anarithmetic unit that simulates behavior of an electromagnetic componentincluding a coil at each of a plurality of time points based on ananalytic model of the electromagnetic component, the arithmetic unitcreating a look-up table storing a flux linkage of the coil, aninductance of the coil and a current in the coil that are obtained by amagnetic field analysis based on the analytic model in association withone another; and simulating behavior of the electromagnetic component byreferring to the look-up table using currents in the coil calculated ata previous simulation step and at a step before the previous simulationstep.

According to the present disclosure, it is possible to simulate thedynamic behavior of an electromagnetic component by solving nonlinearvoltage equations without performing repetitive calculation.

The above and further objects and features will more fully be apparentfrom the following detailed description with accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram illustrating the configuration of a simulationdevice according to Embodiment 1 of the present disclosure.

FIG. 2 is a schematic view illustrating a motor viewed from thedirection of a rotation shaft.

FIG. 3 is a schematic view illustrating the circuit configuration of themotor.

FIG. 4 is a conceptual diagram showing the outline of a coupled analysisto be executed by the simulation device.

FIG. 5 is a flowchart showing a processing procedure relating tocreation of an LUT to be performed by an arithmetic unit.

FIG. 6 is a flowchart showing a processing procedure relating to acoupled analysis to be performed by the arithmetic unit.

FIG. 7 is a flowchart showing a processing procedure relating to a motorbehavior simulation according to Embodiment 1 to be performed by thearithmetic unit.

FIG. 8A is a graph showing a simulation result of phase currentsindicating the work and effect of the simulation device according toEmbodiment 1.

FIG. 8B is a graph showing a simulation result of phase currentsindicating the work and effect of the simulation device according toEmbodiment 1.

FIG. 9A is a graph showing a simulation result of flux linkagesindicating the work and effect of the simulation device according toEmbodiment 1.

FIG. 9B is a graph showing a simulation result of flux linkagesindicating the work and effect of the simulation device according toEmbodiment 1.

FIG. 10A is a graph showing a simulation result of induced voltagesindicating the work and effect of the simulation device according toEmbodiment 1.

FIG. 10B is a graph showing a simulation result of induced voltagesindicating the work and effect of the simulation device according toEmbodiment 1.

FIG. 11A is a graph showing a simulation result of a neutral-pointvoltage indicating the work and effect of the simulation deviceaccording to Embodiment 1.

FIG. 11B is a graph showing a simulation result of a neutral-pointvoltage indicating the work and effect of the simulation deviceaccording to Embodiment 1.

DETAILED DESCRIPTION

The present disclosure will be described below with reference to thedrawings showing embodiments thereof.

Embodiment 1

FIG. 1 is a block diagram illustrating the configuration of a simulationdevice 1 according to Embodiment 1 of the present disclosure. Asimulation device according to an embodiment of the present disclosureis denoted by the reference numeral 1 in the drawing. The simulationdevice 1 is a computer including an arithmetic unit 11 such as a centralprocessing unit (CPU), for example, to which a storage unit 12 isconnected through a bus. The storage unit 12 is provided with anonvolatile memory and a volatile memory, for example. The nonvolatilememory is a ROM such as an electrically erasable programmable ROM or thelike. The nonvolatile memory stores a control program needed for initialoperation of the computer and a simulator program 21 according to thepresent embodiment. The simulator program 21 includes, for example, amotor behavior simulator program (computer program) 21 a, a drivingcircuit simulator program 21 b, a magnetic field analysis simulatorprogram 21 c and so on. By executing the simulator program 21, thearithmetic unit 11 functions as a motor behavior simulator forsimulating the behavior of the motor 4 (see FIG. 2) at each of themultiple time points, as a driving circuit simulator for simulating thebehavior of the driving circuit for driving the motor 4, and as amagnetic field analysis simulator for performing a magnetic fieldanalysis on the behavior of the motor 4 by the magnetic field analysismethod such as a finite-element method or a boundary element method. Thevolatile memory is a RAM such as a dynamic RAM (DRAM), a static RAM(SRAM) or the like, and temporarily stores the control program or thesimulator program 21 read from the nonvolatile memory when thearithmetic processing by the arithmetic unit 11 is executed, or variousdata generated through the arithmetic processing performed by thearithmetic unit 11.

The storage unit 12 further stores an analytic model 12 a representing atwo-dimensional or a three-dimensional shape of multiple coils 42, astator 41 and a rotor 43 (see FIG. 2) that form the motor 4 and theelectromagnetic characteristic thereof, a driving circuit model fordriving a motor 4 and so on.

FIG. 2 is a schematic view illustrating the motor 4 viewed from thedirection of a rotation shaft while FIG. 3 is a schematic viewillustrating the circuit configuration of the motor 4. The motor 4 to besimulated is a three-phase permanent magnet synchronous motor, forexample. The motor 4 illustrated in FIG. 2 is an interior permanentmagnet synchronous motor (PMSM) having eight poles and 48 slots. Themotor 4 includes a cylindrical stator 41 having U phase coils 42 u, Vphase coils 42 v and W phase coils 42 w for generating field flux thatare circumferentially disposed at equally spaced intervals and a rotor43 that is concentrically disposed in the inner diameter side of thestator 41. The coils 42 are star-connected as illustrated in FIG. 3, forexample. In FIG. 3, Tn is the neutral point. Tu, Tv and Tw are terminalsfor applying voltage to the U phase coil 42 u, the V phase coil 42 v andthe w phase coil 42 w, respectively. The rotor 43 is cylindrical andincludes multiple pairs of permanent magnets. It is noted that thenumber of poles, the number of slots and the number of coils 42 are notlimited thereto. The analytic model 12 a includes a three-dimensionalshape model such as three-dimensional CAD data representing the shapesof the multiple coils 42, the stator 41 and the rotor 42 that form themotor 4, for example, and material characteristics or the like of thecomponents that form the three-dimensional shape model. The materialcharacteristics include a magnetizing property, an electriccharacteristic, a mechanical property, a heat characteristic, an ironloss characteristic or the like. The electric characteristics includeconductivity, relative dielectric constant, etc.

The driving circuit to be simulated is formed by a driver and aninverter, for example. The storage unit 12 stores a driving circuitmodel representing multiple circuit elements forming the driver and theinverter, the connected state of the circuit elements and the propertiesof the circuit elements.

Furthermore, the storage unit 12 stores an LUT 12 b and a torque LUT 12c as characteristic databases for simulating the dynamic behavior of themotor 4. Each of the characteristic databases is created beforesimulation of the behavior of the motor 4. The details of the LUT 12 band the torque LUT 12 c will be described later.

It is noted the storage unit 12 may include a readable disk drive suchas a hard disk drive or a solid state drive and a CD-ROM drive or thelike capable of reading data from a portable recording medium 2. Thesimulator program 21 or the motor behavior simulator program 21 aaccording to the present embodiment are computer-readably recorded inthe recording medium 2 such as a compact disc (CD)-ROM, a digitalversatile disc (DVD)-ROM, a blu-ray disk (BD) (Registered trademark) orthe like as portable media. It is noted that the optical disk is oneexample of the recording medium 2. The simulator program 21 or the motorbehavior simulator program 21 a may also be computer-readably recordedin a flexible disk, a magneto-optical disk, an external hard disk, asemiconductor memory or the like. The arithmetic unit 11 reads thesimulator program 21 or the motor behavior simulator program 21 a fromthe recording medium 2 and stores it in the hard disk drive, the solidstate drive or the like. The arithmetic unit 11 executes the simulatorprogram 21 recorded in the recording medium 2 or the simulator program21 stored in the storage unit 12 to cause the computer to function asthe simulation device 1.

Moreover, the simulation device 1 is provided with an input device 13such as a keyboard, a mouse or the like and an output device 14 such asa liquid crystal display, a CRT display or the like and acceptsoperation by the user such as data input.

Additionally, the simulation device 1 is provided with a communicationinterface 15 and may be configured to download the simulator program 21or the motor behavior simulator program 21 a according to the presentdisclosure from an external server computer 3 connected to thecommunication interface 15 and execute the processing by the arithmeticunit 11.

FIG. 4 is a conceptual diagram illustrating the outline of a coupledanalysis to be executed by the simulation device 1. First, thesimulation device 1 calculates various characteristics of the motor 4 bythe magnetic field analysis based on the analytic model 12 a such as afinite-element method model or the like before simulating the behaviorof the motor 4. For example, the arithmetic unit 11 creates the LUT (Ψ,L, I and θ) 12 b, etc. storing, as characteristics of the motor 4, atotal magnetic flux Ψ (hereinafter simply referred to as a flux linkage)being a sum of flux linkage components related to the permanent magnets43 a of the rotor 43 and flux linkage components related to phasecurrents, inductances L of the respective coils 42, currents I flowingin the respective coils 42 and a mechanical angle θ of the rotor 43 inassociation with one another. The inductance L includes aself-inductance of each of the coils 42 and a mutual inductance betweenthe coils 42. The arithmetic unit 11 further creates the torque LUT 12 cstoring a torque T produced in the rotor 43, currents I flowing in therespective coils 42 and the mechanical angle θ of the rotor 43 inassociation with one another.

The simulation device 1 couples the motor behavior simulator with thedriving circuit simulator to thereby simulate the dynamic behavior ofthe motor 4. The driving circuit simulator passes voltages [V]=[Vu, Vvand Vw] to be applied to the terminals Tu, Tv and Tw of the respectivecoils 42 of the motor 4 and the mechanical angle of the rotor 43 to themotor behavior simulator. The motor behavior simulator evaluatescurrents [I]=[Iu, Iv and Iw] of the respective coils 42 and a torque ofthe motor 4 by referring to the LUT 12 b and the torque LUT 12 c usingthe voltages [V] and the mechanical angle or the like of the rotor 43,and returns the simulation results to the driving circuit simulator.From this point onward, by repetitively executing similar processing,the dynamic behavior of the motor 4 can be simulated.

The procedure of creating the LUT 12 b and the procedure of simulatingthe behavior of the motor 4 will be described in order as a simulationmethod according to the present embodiment.

FIG. 5 is a flowchart showing a processing procedure relating tocreation of the LUT 12 b to be performed by the arithmetic unit 11. Thearithmetic unit 11 of the simulation device 1 executes the followingprocessing according to the motor behavior simulator program 21 a storedin the storage unit 12. The arithmetic unit 11 first accepts selectionof the analytic model 12 a and the driving circuit model of the motor 4to be simulated and other various settings by the input device 13 (stepS11).

Then, the arithmetic unit 11 executes a magnetic field analysis by thefinite-element method while setting parameters indicating the drivingstates, that is, setting slightly changed values of currents flowing inthe respective coils 42 and the mechanical angles of the rotor 43 (stepS12). It is noted that when setting values to currents flowing in therespective coils 42, the values of the current flowing in the coils 42are set so as to satisfy the principle of current conservation. In thefinite-element method, the three-dimensional shape model of the motor 4is divided into multiple elements. For example, the arithmetic unit 11divides the three-dimensional shape model of the motor 4 into multipletetrahedral elements, hexahedral elements, quadrangular pyramidelements, triangular prism elements and so on. The arithmetic unit 11calculates the plural simultaneous linear equations obtained from theMaxwell equation under a specific boundary condition, for example, theDirichlet boundary condition or the Neumann boundary condition tothereby evaluate a magnetic vector potential for each of the elements.The magnetic field or the magnetic flux density for each of thecomponents of the motor 4 can be obtained from the magnetic vectorpotential. The magnetic field or the magnetic flux density is basicinformation for calculating a current, a torque or the like. It is notedthat a quasi-stationary magnetic field is described by the Maxwellequation.

Subsequently, the arithmetic unit 11 calculates the flux linkages of therespective coils 42 depending on the currents of the respective coils 42and the position of the rotor 43 based on the magnetic field analysisresults at step S12 (step S13). The arithmetic unit 11 furthercalculates the self-inductances and the mutual inductances of therespective coils 42 (step S14).

The self-inductance and mutual inductance calculated here aredifferential inductances and expressed by Equation (5) below(hereinafter, simply referred to as self-inductance and mutualinductance.).

L=∂Ψ/∂I={Ψ(I+ΔI, θ)−Ψ(I, θ)}/ΔI   (5)

where L is differential inductance (self-inductance and mutualinductance)

-   Ψ is the flux linkage of each coil 42,-   I is current flowing in each coil 42-   ΔI is minute variation of current.

It is noted that, for a three-phase coil, the above-describeddifferential inductances are evaluated for each of the flux linkages ofthe coils 42 and for each of the slightly varying currents flowing inthe coils 42, and are represented by a 3-by-3 inductance matrix. The 9matrix components arranged in a 3-by-3 matrix are expressed asLuu=∂Ψu/∂Iu, Luv=∂Ψu/∂Iv, Luw=∂Ψu/∂Iw, Lvu=∂Ψv/∂Iu, Lvv=∂Ψv/∂Iv,Lvw=∂Ψv/∂Iw, Lwu=∂Ψw/∂Iu, Lwv=∂Ψw/∂Iv, and Lww=∂Ψw/∂Iw.

Then, the arithmetic unit 11 calculates an electromagnetic force exertedon the rotor 43 depending on the currents flowing in the respectivecoils 42 and the position of the rotor 43 based on the magnetic fieldanalysis results at step S12 to calculate a torque exerted on the rotor43 (step S15). The arithmetic unit 11 calculates the electromagneticforce exerted on the rotor 43 by using a nodal force method, forexample.

Next, the arithmetic unit 11 creates the LUT 12 b storing the fluxlinkages of the respective coils, the self-inductances and the mutualinductances of the respective coils 42, currents flowing in therespective coils 42 and the mechanical angle of the rotor 43 inassociation with one another, based on the magnetic field analysisresults at step S12 (step S16).

Then, the arithmetic unit 11 creates the torque LUT 12 c storing thetorque calculated at step S15, the current flowing in the respectivecoils 42 and the mechanical angle of the rotor 43 in association withone another (step S17) and ends the processing.

It is noted that though the LUT 12 b and the torque LUT 12 c aredescribed as separate tables, these tables may be configured as a singletable.

FIG. 6 is a flowchart showing a processing procedure relating to acoupled analysis to be performed by the arithmetic unit 11. Thearithmetic unit 11 sets initial values of voltages to be applied to thecoils 42, currents therein, the position of the rotor 43, etc. (stepS31).

Then, the arithmetic unit 11 calculates voltages to be applied to thecoils 42 and a mechanical angle of the rotor 43 at the next simulationstep based on the currents of the respective coils 42 and the torqueexerted on the rotor 43 calculated at the previous simulation step (stepS32). The processing at step S32 is executed by the driving circuitsimulator (see FIG. 4) and provides the motor behavior simulator withthe voltages [V]=[Vu, Vv, Vw] and the mechanical angle of the rotor 43as the simulation results.

Subsequently, the arithmetic unit 11 simulates the behavior of the motor4 based on the voltages to be applied to the motor 4, and the currentsof the respective coils 42, the flux linkages of the respective coils 42and the position of the rotor 43 that had been calculated before theprevious simulation step, and calculates currents flowing in therespective coils 42 and a torque produced in the rotor 43 (step S33).The processing at step S33 is executed by the motor behavior simulator(see FIG. 4), and passes the currents [I]=[Iu, Iv, Iw] of the respectivecoils 42 and the torque produced in the rotor 43 as simulation resultsto the driving circuit simulator. The details of the processing at stepS33 will be described below.

Next, the arithmetic unit 11 determines whether or not a simulation endcondition is satisfied (step S34). For example, the arithmetic unit 11ends the simulation if a predetermined number of simulation stepscorresponding to a predetermined real time are executed. If determiningthat the simulation end condition is not satisfied (step S34: NO), thearithmetic unit 11 returns the processing to step S32 to repetitivelyexecute the processing at steps S32 and S33. If determining that thesimulation end condition is satisfied (step S34: YES), the arithmeticprocessing 11 ends the processing.

Before the details of the processing procedure to be performed by thearithmetic unit 11 at step S33 is described, the theory of a method ofcalculating currents flowing in the respective coils 42 and a voltage atthe neutral point based on the voltages to be applied to the respectivecoils 42 will be described below.

The voltage equations of the star-connected circuit as illustrated inFIG. 3 can be expressed by Equations (1)-(3) above. Moreover, in theneutral point Tn, the principle of current conservation as expressed byEquation (4) above is established.

Taking note of the voltage equation of the U phase coil 42 u,substituting the time differential Δt for the time differential of thevoltage equation expressed by Equation (1) above gives Equation (6)below.

$\begin{matrix}{\mspace{79mu}{{{V_{u\; 0}^{(n)} - V_{n}^{(n)}} = {{RI}_{u}^{(n)} + \frac{{\Psi_{u}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{u}\left( {\text{?},\theta^{({n - 1})}} \right)}}{\Delta\; t}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (6)\end{matrix}$

Note that the upper-right superscript (n) means a value at the n-thsimulation step.

-   V_(u0) ^((n)): voltage to be applied to the terminal of the U phase    coil-   V_(n) ^((n)): voltage at the neutral point-   I    ^((n)): (I_(u) ^((n)), I_(v) ^((n)), I_(w) ^((n)))-   I_(u) ^((n)) I_(v) ^((n)), I_(w) ^((n)): current timing in each coil-   θ^((n)): mechanical angle of the rotor-   Ψ_(u)(I    ^((n)), θ^((n))): flux linkage in the U phase coil-   Δt: time increment between simulation steps

Here, the flux linkage Ψ is dependent on the current I of an unknown,and can be explicitly expressed by using the current I The flux linkagecomponent derived from current can be described by using a differentialinductance and a current, and thus the time differential component ofthe flux linkage in Equation (6) above is separated into the fluxlinkage component derived from current and the rest, and can beexpressed by Equation (7) below.

$\begin{matrix}{{\frac{{\Psi_{u}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{u}\left( {\text{?},\theta^{({n - 1})}} \right)}}{\Delta\; t} = {\left( {{{L_{uu}\left( {\text{?},\theta^{(n)}} \right)}\frac{I_{u}^{(n)} - I_{u}^{({n - 1})}}{\Delta\; t}} + {{L_{uv}\left( {\text{?},\theta^{(n)}} \right)}\frac{I_{v}^{(n)} - I_{v}^{({n - 1})}}{\Delta\; t}} + {{L_{uw}\left( {\text{?},\theta^{(n)}} \right)}\frac{I_{w}^{(n)} - I_{w}^{({n - 1})}}{\Delta\; t}}} \right) + \frac{{\Psi_{u}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{u}\left( {\text{?},\theta^{({n - 1})}} \right)}}{\Delta\; t} - \left( {{{L_{uu}\left( {\text{?},\theta^{(n)}} \right)}\frac{\text{?} - \text{?}}{\Delta\; t}} + {{L_{uv}\left( {\text{?},\theta^{(n)}} \right)}\frac{I_{v}^{(n)} - I_{v}^{({n - 1})}}{\Delta\; t}} + {{L_{uw}\left( {\text{?},\theta^{(n)}} \right)}\frac{\text{?} - \text{?}}{\Delta\; t}}} \right)}}{\text{?}\text{indicates text missing or illegible when filed}}} & (7)\end{matrix}$

where L_(uu)(I

^((n)), θ^((n))): self-inductance of the U phase coil

-   -   L_(uv)(I        ^((n)), θ^((n))): mutual inductance between the U phase coil and        the V phase coil    -   L_(uw)(I        ^((n)), θ^((n))): mutual inductance between the U phase coil and        the W phase coil

The second term and the third term on the right side of Equation (7)above represent voltages caused by a factor other than change incurrent, such as rotation of the rotor 43 or the like. When the secondterm and the third term are further approximated to values at theprevious simulation step, Equation (7) above can be expressed byEquation (8) below.

$\begin{matrix}{{\frac{{\Psi_{u}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{u}\left( {\text{?},\theta^{({n - 1})}} \right)}}{\Delta\; t} \approx {\left( {{{L_{uu}\left( {\text{?},\theta^{(n)}} \right)}\frac{I_{u}^{(n)} - I_{u}^{({n - 1})}}{\Delta\; t}} + {{L_{uv}\left( {\text{?},\theta^{(n)}} \right)}\frac{I_{v}^{(n)} - I_{v}^{({n - 1})}}{\Delta\; t}} + {{L_{uw}\left( {\text{?},\theta^{(n)}} \right)}\frac{I_{w}^{(n)} - I_{w}^{({n - 1})}}{\Delta\; t}}} \right) + \frac{{\Psi_{u}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{u}\left( {\text{?},\theta^{({n - 1})}} \right)}}{\Delta\; t} + {- \left( {{{L_{uu}\left( {\text{?},\theta^{({n - 1})}} \right)}\frac{I_{u}^{({n - 1})} - I_{u}^{({n - 2})}}{\Delta\; t}} + {{L_{uv}\left( {\text{?},\theta^{({n - 1})}} \right)}\frac{I_{v}^{({n - 1})} - I_{v}^{({n - 2})}}{\Delta\; t}} + {{L_{uw}\left( {\text{?},\theta^{({n - 1})}} \right)}\frac{I_{w}^{({n - 1})} - I_{w}^{({n - 2})}}{\Delta\; t}}} \right)}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (8)\end{matrix}$

Substituting Equation (8) above into Equation (6) above gives Equation(9) below.

$\begin{matrix}{{{V_{u\; 0}^{(n)} - V_{n}^{(n)}} = {{RI}_{u}^{(n)} + \left( {{{L_{uu}\left( {\text{?},\theta^{(n)}} \right)}\frac{I_{u}^{(n)} - I_{u}^{({n - 1})}}{\Delta\; t}} + {{L_{uv}\left( {\text{?},\theta^{(n)}} \right)}\frac{I_{v}^{(n)} - I_{v}^{({n - 1})}}{\Delta\; t}} + {{L_{uw}\left( {\text{?},\theta^{(n)}} \right)}\frac{I_{w}^{(n)} - I_{w}^{({n - 1})}}{\Delta\; t}}} \right) + \frac{{\Psi_{u}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{u}\left( {\text{?},\theta^{({n - 1})}} \right)}}{\Delta\; t} + {- \left( {{{L_{uu}\left( {\text{?},\theta^{({n - 1})}} \right)}\frac{I_{u}^{({n - 1})} - I_{u}^{({n - 2})}}{\Delta\; t}} + {{L_{uv}\left( {\text{?},\theta^{({n - 1})}} \right)}\frac{I_{v}^{({n - 1})} - I_{v}^{({n - 2})}}{\Delta\; t}} + {{L_{uw}\left( {\text{?},\theta^{({n - 1})}} \right)}\frac{I_{w}^{({n - 1})} - I_{w}^{({n - 2})}}{\Delta\; t}}} \right)}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (9)\end{matrix}$

Multiplying both sides of Equation (9) above by time Δt, and transposingthe current I of the unknown and the voltage Vn of the unknown in theneutral point Tn to the left side and transposing the known term to theright side give Equation (10) below.

$\begin{matrix}{{{{{RI}_{u}^{(n)}\Delta\; t} + {{L_{uu}\left( {\text{?},\theta^{(n)}} \right)}I_{u}^{(n)}} + {{L_{uv}\left( {\text{?},\theta^{(n)}} \right)}I_{v}^{(n)}} + {{L_{uw}\left( {\text{?},\theta^{(n)}} \right)}I_{w}^{(n)}} + {V_{n}^{(n)}\Delta\; t}} = {{V_{u\; 0}^{(n)}\Delta\; t} + \left( {{{L_{uu}\left( {\text{?},\theta^{(n)}} \right)}\text{?}} + {{L_{uv}\left( {\text{?},\theta^{(n)}} \right)}I_{v}^{({n - 1})}} + {{L_{uw}\left( {\text{?},\theta^{(n)}} \right)}I_{w}^{({n - 1})}}} \right) + \left( {{{L_{uu}\left( {\text{?},\theta^{({n - 1})}} \right)}\left( {I_{u}^{({n - 1})} - I_{u}^{({n - 2})}} \right)} + {{L_{uv}\left( {\text{?},\theta^{({n - 1})}} \right)}\left( {I_{v}^{({n - 1})} - I_{v}^{({n - 2})}} \right)} + {{L_{uw}\left( {\text{?},\theta^{({n - 1})}} \right)}\left( {I_{w}^{({n - 1})} - I_{w}^{({n - 2})}} \right)}} \right) - \left( {{\Psi_{u}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{u}\left( {\text{?},\theta^{({n - 1})}} \right)}} \right)}}{\text{?}\text{indicates text missing or illegible when filed}}} & (10)\end{matrix}$

Similarly to the U phase coil 42 u, the voltage equation of the V phasecoil 42 v expressed by Equation (2) above is expressed by Equation (11)below.

$\begin{matrix}{{{{{RI}_{v}^{(n)}\Delta\; t} + {{L_{vu}\left( {\text{?}^{({n - 1})},\theta^{(n)}} \right)}I_{u}^{(n)}} + {{L_{vv}\left( {\text{?}^{({n - 1})},\theta^{(n)}} \right)}I_{v}^{(n)}} + {{L_{vw}\left( {\text{?}^{({n - 1})},\theta^{(n)}} \right)}I_{w}^{(n)}} + {V_{n}^{(n)}\Delta\; t}} = {{V_{v0}^{(n)}\Delta\; t} + \left( {{{L_{vu}\left( {\text{?}^{({n - 1})},\theta^{(n)}} \right)}I_{u}^{({n - 1})}} + {{L_{vv}\left( {\text{?}^{({n - 1})},\theta^{(n)}} \right)}I_{v}^{({n - 1})}} + {{L_{vw}\left( {\text{?}^{({n - 1})},\theta^{(n)}} \right)}I_{w}^{({n - 1})}}} \right) + \left( {{{L_{vu}\left( {\text{?}^{({n - 2})},\theta^{({n - 1})}} \right)}\left( {I_{u}^{({n - 1})} - I_{u}^{({n - 2})}} \right)} + {{L_{uv}\left( {\text{?}^{({n - 2})},\theta^{({n - 1})}} \right)}\left( {I_{v}^{({n - 1})} - I_{v}^{({n - 2})}} \right)} + {{L_{vw}\left( {\text{?}^{({n - 2})},\theta^{({n - 1})}} \right)}\left( {I_{w}^{({n - 1})} - I_{w}^{({n - 2})}} \right)}} \right) - \left( {{\Psi_{v}\left( {\text{?}^{({n - 1})},\theta^{(n)}} \right)} - {\Psi_{v}\left( {\text{?}^{({n - 2})},\theta^{({n - 1})}} \right)}} \right)}}{\text{?}\text{indicates text missing or illegible when filed}}} & (11)\end{matrix}$

-   where V_(v0) ^((n)): voltage to be applied to the terminal of the V    phase coil-   L_(vv)(I    ^((n)), θ^((n))): self-inductance of the V phase coil-   L_(vu)(I    ^((n)), θ^((n))): mutual inductance. between the V phase coil and    the U phase coil-   L_(vw)(I    ^((n)), θ^((n))): mutual inductance between the V phase coil and the    W phase coil

Similarly to the U phase coil 42 u, the voltage equation of the W phasecoil 42 w expressed by Equation (3) above is expressed by Equation (12)below.

$\begin{matrix}{{{{{RI}_{w}^{(n)}\Delta\; t} + {{L_{wu}\left( {\text{?},\theta^{(n)}} \right)}I_{u}^{(n)}{L_{wv}\left( {\text{?},\theta^{(n)}} \right)}I_{v}^{(n)}} + {{L_{ww}\left( {\text{?},\theta^{(n)}} \right)}I_{w}^{(n)}} + {V_{n}^{(n)}\Delta\; t}} = {{V_{w\; 0}^{(n)}\Delta\; t} + \left( {{{L_{wu}\left( {\text{?},\theta^{(n)}} \right)}I_{u}^{({n - 1})}} + {{L_{wv}\left( {\text{?},\theta^{(n)}} \right)}I_{v}^{({n - 1})}} + {{L_{ww}\left( {\text{?},\theta^{(n)}} \right)}I_{w}^{({n - 1})}}} \right) + \left( {{{L_{wu}\left( {\text{?},\theta^{({n - 1})}} \right)}\left( {I_{u}^{({n - 1})} - I_{u}^{({n - 2})}} \right)} + {{L_{wv}\left( {\text{?},\theta^{({n - 1})}} \right)}\left( {I_{v}^{({n - 1})} - I_{v}^{({n - 2})}} \right)} + {{L_{ww}\left( {\text{?},\theta^{({n - 1})}} \right)}\left( {I_{w}^{({n - 1})} - I_{w}^{({n - 2})}} \right)}} \right) - \left( {{\Psi_{w}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{w}\left( {\text{?},\theta^{({n - 1})}} \right)}} \right)}}{\text{?}\text{indicates text missing or illegible when filed}}} & (12)\end{matrix}$

where V_(w0) ^((n)): voltage to be applied to the terminal of the Wphase coil

-   L_(ww)(I    ^((n)), θ^((n))): self-inductance of the W phase coil-   L_(wu)(I    ^((n)), θ^((n))): mutual inductance between the W phase coil and the    U phase coil-   L_(wv)(I    ^((n)), θ^((n))): mutual inductance between the W phase coil and the    V phase coil

Equations (10), (11) and (12) above and the equation for conservation ofcurrent in Equation (4) above are collectively described in a matrixform to give Equation (13) below.

$\begin{matrix}{{{\begin{pmatrix}{{R\;\Delta\; t} + {L_{uu}\left( {\text{?},\theta^{(n)}} \right)}} & {L_{uv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{uw}\left( {\text{?},\theta^{(n)}} \right)} & {\Delta\; t} \\{L_{vu}\left( {\text{?},\theta^{(n)}} \right)} & {{R\;\Delta\; t} + {L_{vv}\left( {\text{?},\theta^{(n)}} \right)}} & {L_{vw}\left( {\text{?},\theta^{(n)}} \right)} & {\Delta\; t} \\{L_{wu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{wv}\left( {\text{?},\theta^{(n)}} \right)} & {{R\;\Delta\; t} + {L_{ww}\left( {\text{?},\theta^{(n)}} \right)}} & {\Delta\; t} \\1 & 1 & 1 & 0\end{pmatrix}\begin{pmatrix}I_{u}^{(n)} \\I_{v}^{(n)} \\I_{w}^{(n)} \\V_{n}^{(n)}\end{pmatrix}} = {\begin{pmatrix}{V_{u\; 0}^{(n)}\Delta\; t} \\{V_{v\; 0}^{(n)}\Delta\; t} \\{V_{w\; 0}^{(n)}\Delta\; t} \\0\end{pmatrix} + {\begin{pmatrix}{L_{uu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{uv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{uw}\left( {\text{?},\theta^{(n)}} \right)} & 0 \\{L_{vu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{vv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{vw}\left( {\text{?},\theta^{(n)}} \right)} & 0 \\{L_{wu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{wv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{ww}\left( {\text{?},\theta^{(n)}} \right)} & 0 \\0 & 0 & 0 & 0\end{pmatrix}\begin{pmatrix}I_{u}^{({n - 1})} \\I_{v}^{({n - 1})} \\I_{w}^{({n - 1})} \\0\end{pmatrix}} + {\begin{pmatrix}{L_{uu}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{uv}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{uw}\left( {\text{?},\theta^{({n - 1})}} \right)} & 0 \\{L_{vu}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{vv}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{vw}\left( {\text{?},\theta^{({n - 1})}} \right)} & 0 \\{L_{wu}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{wv}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{ww}\left( {\text{?},\theta^{({n - 1})}} \right)} & 0 \\\; & 0 & 0 & 0\end{pmatrix}\begin{pmatrix}{I_{u}^{({n - 1})} - I_{u}^{({n - 2})}} \\{I_{v}^{({n - 1})} - I_{v}^{({n - 2})}} \\{I_{w}^{({n - 1})} - I_{w}^{({n - 2})}} \\0\end{pmatrix}} - \begin{pmatrix}{{\Psi_{u}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{u}\left( {\text{?},\theta^{({n - 1})}} \right)}} \\{{\Psi_{v}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{v}\left( {\text{?},\theta^{({n - 1})}} \right)}} \\{{\Psi_{w}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{w}\left( {\text{?},\theta^{({n - 1})}} \right)}}\end{pmatrix}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (13)\end{matrix}$

In Equation (13) above, the inductances and the flux linkages can beread by referring to the LUT 12 b using the currents I=(Iu, Iv, Iw) andthe mechanical angle θ of the rotor 43 as keys. According to Equation(13) above, by using the information at the previous simulation step(the (n−1)-th step) and at the step before the previous simulation step(the (n−2)-th step), the currents flowing in the respective coils 42 andthe voltage at the neutral point Tn at the current simulation step (then-th step) can be calculated without performing repetitive calculation.

FIG. 7 is a flowchart showing a processing procedure relating to themotor behavior simulation according to Embodiment 1 to be performed bythe arithmetic unit 11. The processing procedure at step S33 in the n-thstep will be described below. Here, n is an integer equal to or largerthan 3. The arithmetic unit 11 acquires the voltages to be applied tothe respective coils 42 and the mechanical angle of the rotor 43 fromthe driving circuit simulator (step S51). For example, in a case wherethe driving circuit simulator is configured to output the simulationresult in a file format, the arithmetic unit 11 reads out the voltagesto be applied to the respective coils 42 and the mechanical angle of therotor 43 from the file.

The arithmetic unit 11 reads out the self-inductances and the mutualinductances of the respective coils 42 by referring to the LUT 12 busing the currents of the respective coils 42 at the (n−1)-th simulationstep and the mechanical angle of the rotor 43 at the n−th simulationstep as keys (step S52). Furthermore, the arithmetic unit 11 reads outthe self-inductances and the mutual inductances of the respective coils42 by referring to the LUT 12 b using the currents of the respectivecoils 42 at the (n−2)-th simulation step and the mechanical angle of therotor 43 at the (n−1)-th simulation step as keys (step S53).

Then, the arithmetic unit 11 reads out the flux linkages of therespective coils 42 by referring to the LUT 12 b using the currents ofthe respective coils 42 at the (n−1)-th simulation step and themechanical angle of the rotor 43 at the n-th simulation step as keys(step S54). Furthermore, the arithmetic unit 11 reads out the fluxlinkages of the respective coils 42 by referring to the LUT 12 b usingthe currents of the respective coils at the (n−2)-th simulation step andthe mechanical angle of the rotor 43 at the (n−1)-th simulation step askeys (step S55).

For the convenience of explanation, though the processing steps aredescribed from steps S52 to S55 in order, the processing order is notlimited thereto. Moreover, the processing at steps S52 and S54 maysimultaneously be executed. The processing at steps S53 and S55 maysimultaneously be executed.

The arithmetic unit 11 calculates currents flowing in the respectivecoils 42 and a voltage at the neutral point Tn at the n-th simulationstep by Equation (13) above based on the voltages to be applied to therespective coils 42, the inductances and the flux linkages read out atsteps S52-S55 (step S56).

Then, the arithmetic unit 11 reads out the torque exerted on the rotor43 by referring to the torque LUT 12 c using the currents of therespective coils 42 calculated at step S56 and the mechanical angle ofthe rotor 43 as keys (step S57).

The arithmetic unit 11 then outputs the currents of the respective coils42 calculated at step S56 and the torque read out at step S57 to thedriving circuit simulator (step S58) and ends the processing.

The work and effect of the simulation device 1 thus configured, thesimulation method and the simulator program 21 are described. The actualsimulations for demonstrating the work and effect are performed by usingthe interior permanent magnet synchronous motor having a shapeillustrated in FIG. 2 and having eight poles and 48 slots. The coils 42are wound around the slots of the stator 41 by distributed winding.

FIGS. 8A and 8B, FIGS. 9A and 9B, FIGS. 10A and 10B and FIGS. 11A and11B are graphs respectively showing the simulation results of phasecurrents, flux linkages, induced voltages and neutral point voltagesrepresenting the work and effect of the simulation device 1 according toEmbodiment 1. FIG. 8A, FIG. 9A, FIG. 10A and FIG. 11A show arithmeticresults obtained when the behavior of the motor 4 is simulated by usingthe simulation device 1 according to Embodiment 1 while FIG. 8B, FIG.9B, FIG. 10B and FIG. 11B show arithmetic results obtained when thebehavior of the motor 4 is simulated by the magnetic field analysisusing the finite-element method. The horizontal axes of the graphsrepresent time. The vertical axes of the graphs in FIGS. 8A and 8Brepresent currents flowing in the respective coils 42, the vertical axesof the graphs in FIGS. 9A and 9B represent flux linkages of therespective coils 42, the vertical axes of the graphs in FIGS. 10A and10B represent induced voltages of the respective coils 42, and thevertical axes of the graphs in FIGS. 11A and 11B represent voltages inthe neutral point.

As illustrated in FIGS. 8A and 8B to FIGS. 11A and 11B, the phasecurrents, flux linkages, induced voltages and neutral point voltage ofthe respective coils 42 are reproduced with high accuracy.

As described above, the simulation device 1, the simulation method andthe simulator program 21 according to Embodiment 1 can simulate thedynamic behavior of the motor 4 by solving the non-linear voltageequations without performing repetitive calculation.

Furthermore, as illustrated in FIGS. 8A and 8B to FIGS. 11A and 11B, thesimulation results substantially the same as the arithmetic results thatare strict solutions obtained by the magnetic field analysis can beobtained, which enables accurate simulation of the behavior of the motor4.

In the present embodiment, though the present embodiment described themotor 4 as a rotary machine in which a mover is rotated, the presentinvention can be applied to a motor 4 acting as a linear motion machinein which a mover linearly moves to thereby simulate the dynamic behaviorof the motor. This simulation of the behavior of the linear motionmachine can be performed by a similar processing procedure with a slightdifference in shape of the analytic model 12 a.

Furthermore, though the present embodiment described an object to beanalyzed that allows a mover to linearly move or rotatably move, themoving manner of the mover is not limited thereto. The present inventioncan also be applied to a motor allowing a mover to vibrate, a linearmotor or a solenoid actuator allowing a mover to linearly move, or thelike. Moreover, the present invention can be applied to an inductionmachine as well.

Additionally, the applicable object of the present invention is notlimited to simulation of a motor having a mover, and the presentinvention can be applied to any electromagnetic component includingmultiple coils. For example, the present invention can also be appliedto the case where the behavior of a motionless machine such as atransformer, a contactless charger or the like is simulated.

Furthermore, in the present embodiment, an example is described in whichvoltages are passed from the driving circuit simulator to the motorbehavior simulator while currents and a torque are returned from themotor behavior simulator to the driving circuit simulator. However, thephysical quantities to be exchanged between the respective simulatorsare not limited thereto and may appropriately be selected. Moreover,physical constants representing the states of the motor 4 or a generatormay be exchanged.

For example, currents are passed from the driving circuit simulator tothe motor behavior simulator while voltages are returned from the motorbehavior simulator to the driving circuit simulator. In this case, themotor behavior simulator solves Equation 13 described above assumingthat the currents are known quantities while the voltage of therespective coils 42 are unknown quantities to thereby evaluate voltagesgenerated in the respective coils 42, and returns the evaluated voltagesto the driving circuit simulator.

In addition, in Embodiment 1, the LUT 12 b is described as informationstoring the currents flowing in the respective coils 42, the mechanicalangle of the rotor 43, the flux linkages of the respective coils 42, andthe inductances of the respective coils 42 in association with oneanother. Alternatively, it may be possible to use a first LUT thatoutputs the self-inductances and the mutual inductances of therespective coils 42 if the currents of the respective coils 42 and themechanical angle of the rotor 43 are input and a second LUT that outputsthe flux linkages of the respective coils 42 if the current of therespective coils 42 and the mechanical angle of the rotor 43 are input.

Furthermore, though the simulation of the behavior of the star-connectedmotor 4 is mainly described in Embodiment 1, the present invention canbe applied to any circuit, and the connection method is not limitedthereto. Thus, the present invention can also be applied to the casewhere the behavior of the motor 4 employing any connection method suchas a delta connection is simulated. The delta connection is similar in abasic concept to that of the star connection, and the voltage equationis expressed by Equation (14) below. By solving Equation (14) below in aprocedure similar to Embodiment 1, the behavior of the motor 4 can besimulated.

$\begin{matrix}{{{\begin{pmatrix}{{R\;\Delta\; t} + {L_{uu}\left( {\text{?},\theta^{(n)}} \right)}} & {L_{uv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{uw}\left( {\text{?},\theta^{(n)}} \right)} \\{L_{vu}\left( {\text{?},\theta^{(n)}} \right)} & {{R\;\Delta\; t} + {L_{vv}\left( {\text{?},\theta^{(n)}} \right)}} & {L_{vw}\left( {\text{?},\theta^{(n)}} \right)} \\{L_{wu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{wv}\left( {\text{?},\theta^{(n)}} \right)} & {{R\;\Delta\; t} + {L_{ww}\left( {\text{?},\theta^{(n)}} \right)}}\end{pmatrix}\begin{pmatrix}I_{u}^{(n)} \\I_{v}^{(n)} \\I_{w}^{(n)}\end{pmatrix}} = {\begin{pmatrix}{V_{u\; 0}^{(n)} - {V_{v\; 0}^{(n)}\Delta\; t}} \\{V_{v\; 0}^{(n)} - {V_{w\; 0}^{(n)}\Delta\; t}} \\{V_{w\; 0}^{(n)} - {V_{u\; 0}^{(n)}\Delta\; t}}\end{pmatrix} + {\begin{pmatrix}{L_{uu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{uv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{uw}\left( {\text{?},\theta^{(n)}} \right)} \\{L_{vu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{vv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{vw}\left( {\text{?},\theta^{(n)}} \right)} \\{L_{wu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{wv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{ww}\left( {\text{?},\theta^{(n)}} \right)}\end{pmatrix}\begin{pmatrix}I_{u}^{({n - 1})} \\I_{v}^{({n - 1})} \\I_{w}^{({n - 1})}\end{pmatrix}} + {\begin{pmatrix}{L_{uu}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{uv}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{uw}\left( {\text{?},\theta^{({n - 1})}} \right)} \\{L_{vu}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{vv}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{vw}\left( {\text{?},\theta^{({n - 1})}} \right)} \\{L_{wu}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{wv}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{ww}\left( {\text{?},\theta^{({n - 1})}} \right)}\end{pmatrix}\begin{pmatrix}{I_{u}^{({n - 1})} - I_{u}^{({n - 2})}} \\{I_{v}^{({n - 1})} - I_{v}^{({n - 2})}} \\{I_{w}^{({n - 1})} - I_{w}^{({n - 2})}}\end{pmatrix}} - \begin{pmatrix}{{\Psi_{u}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{u}\left( {\text{?},\theta^{({n - 1})}} \right)}} \\{{\Psi_{v}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{v}\left( {\text{?},\theta^{({n - 1})}} \right)}} \\{{\Psi_{w}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{w}\left( {\text{?},\theta^{({n - 1})}} \right)}}\end{pmatrix}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (14)\end{matrix}$

Additionally, the behavior of the motor 4 can be simulated for adisconnected circuit as well as a normal circuit. For example, if thew-phase coil is disconnected to cause lack of a W phase, Equation (13)or (14) is solved assuming that Iw is equal to zero, whereby thebehavior of the motor 4 in the disconnected state can be simulated.

Embodiment 2

A simulation device 1, a simulation method and a simulator program 21according to Embodiment 2 are different from those of Embodiment 1 inthat the behavior of an x-phase motor including a stator and a mover issimulated, and thus only the difference mainly will be described below.Here, X is a natural number equal to or larger than four. In thefollowing description in Embodiment 2, the mover is assumed not to belimited to the rotor 43. Furthermore, it is assumed that the electricalresistances of the coils are not necessarily the same in the followingdescription. Embodiment 2 is similar to Embodiment 1 in theconfiguration and work and effect otherwise, and thus the detaileddescription will not be made by applying similar reference symbols tothe corresponding parts.

The voltage equations of the x phase motor can be expressed by Equation(15) below based on a concept similar to Embodiment 1.

$\begin{matrix}{{{\begin{pmatrix}{{L\left( {\text{?},\theta^{(n)}} \right)} + {R\;\Delta\; t}} & {1\Delta\; t} \\1^{T} & 0\end{pmatrix}\begin{pmatrix}I^{(n)} \\V_{n}^{(n)}\end{pmatrix}} = {\begin{pmatrix}{V_{0}^{(n)}\Delta\; t} \\0\end{pmatrix} + {\begin{pmatrix}{L\left( {\text{?},\theta^{(n)}} \right)} & 0 \\0 & 0\end{pmatrix}\begin{pmatrix}I^{({n - 1})} \\0\end{pmatrix}} + {\begin{pmatrix}{L\left( {\text{?},\theta^{({n - 1})}} \right)} & 0 \\0 & 0\end{pmatrix}\begin{pmatrix}{I^{({n - 1})} - I^{({n - 2})}} \\0\end{pmatrix}} - \begin{pmatrix}{{\Psi\left( {\text{?},\theta^{(n)}} \right)} - {\Psi\left( {\text{?},\theta^{({n - 1})}} \right)}} \\0\end{pmatrix}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (15)\end{matrix}$

where the upper-right superscript (n) means a value at the n-thsimulation step.

(L(?^((n)), θ^((n))))_(ij) = L_(ij)(?^((n)), θ^((n)))(R)_(ij) = R_(i)δ_(ij) 1 = (1, 1 . . . 1)^(T) : x − order  unit  vectorI^((n)) = (I₁^((n)), I₂^((n)), . . . , I_(x)^((n)))^(T)V₀^((n)) = (V₁₀^((n)), V₂₀^((n)), . . . , V_(x0)^((n)))^(T)Ψ(?^((n)), θ^((n))) = (Ψ₁(?^((n)), θ^((n))), Ψ₂(?^((n)), θ^((n))), . . . , Ψ_(x)(?^((n)), θ^((n))))^(T)?indicates text missing or illegible when filed

-   -   if: 1, 2, . . . , χ    -   χ: the number of phases for the motor    -   R        : electrical resistance of each coil    -   Δt: time increment between simulation steps    -   δ_(ij): Kronecker's delta    -   L_(ij)(I        ^((n)), θ^((n))): self-inductance (i=j) of the i phase coil or        mutual inductance (i≠j) between the i phase coil and the j phase        coil    -   I        ^((n)): (I₁ ^((n)), I₂ ^((n)), . . . , I_(χ) ^((n)))    -   θ^((n)): position of the mover    -   V_(n) ^((n)): voltage at the neutral point    -   I₁ ^((n)), I₂ ^((n)), . . . , I₁₀₂ ^((n)): current flowing in        each coil    -   V₁₀ ^((n)), V₂₀ ^((n)), . . . , V_(χ0) ^((n)): voltage to be        applied to each coil    -   Ψ₁, Ψ₂, . . . , Ψ_(χ): flux linkage in each coil

The LUT 12 b is a table storing the currents flowing in the respectivecoils 42 of the χ phase, the position of the mover, the self-inductancesand the mutual inductances of the respective coils 42 and the fluxlinkages of the respective coils 42 in association with one another.

The arithmetic unit 11 reads out the self-inductances and the mutualinductances of the respective coils 42 and the flux linkages of therespective coils 42 by referring to the LUT 12 b using the currents ofthe respective coils 42 at the (n−1)-th simulation step and themechanical angle of the mover at the n-th simulation step as keyssimilarly to Embodiment 1. Furthermore, the arithmetic unit 11 reads outthe self-inductances and the mutual inductances of the respective coils42 and the flux linkages of the respective coils 42 by referring to theLUT 12 b using the currents of the respective coils 42 at the (n−2)-thsimulation step and the mechanical angle of the rotor 43 at the (n−1)-thsimulation step as keys.

The arithmetic unit 11 calculates current flowing in the respectivecoils 42 and a voltage at the neutral point Tn at the n-th simulationstep by Equation (15) above based on the voltages to be applied to therespective coils 42, the inductances and the flux linkages read out atthe above-described processing.

Likewise, the arithmetic unit 11 executes processing of reading out thetorque exerted on the rotor 43 from the torque LUT 12 c using thecurrents of the respective coils 42 obtained through the calculation andthe mechanical angle of the rotor 43 as keys and outputting the currentsof the respective coils 42 obtained through the calculation and thetorque read out to the driving circuit simulator.

As described above, according to the simulation device 1, the simulationmethod and the simulator program 21 in Embodiment 2, it is possible tosimulate the dynamic behavior of the motor 4 by solving the nonlinearvoltage equations without performing repetitive calculation similarly toEmbodiment 1.

Though Embodiment 2 described the star-connected χ-phase motor used asone example, the connection method is not particularly limited theretoin the χ-phase motor as well.

Embodiment 3

A simulation device 1, a simulation method and a simulator program 21according to Embodiment 3 are different from Embodiments 1 and 2 in thatthe behavior of a transformer having χ pieces of coils 42 is simulated,and thus only the difference will be described below. Here, χ is anatural number equal to or larger than 1. The χ piece or pieces of coils42 are each primary coils, for example. If there are multiple coils 42,the following description is made assuming that the electricalresistances of the coils 42 are not necessarily the same. Embodiment 3is similar to Embodiment 1 in the configuration and effect otherwise,and thus the detailed description will not be made here by applyingsimilar reference symbols to the corresponding parts.

The voltage equations of the respective coils 42 are similar to Equation(15) above in Embodiment 2, and an equation independent of themechanical angle θ is conceivable. In the case of a transformer failingto have the neutral point, an equation from which the neutral pointvoltage is neglected may be conceived.

The processing procedure performed by the arithmetic unit 11 is similarto that of Embodiment 2 except that a mover is not present and a torqueis not calculated.

As described above, according to the simulation device 1, the simulationmethod and the simulator program 21 in Embodiment 3, it is possible tosimulate the dynamic behavior of the transformer by solving thenonlinear voltage equations without performing repetitive calculationsimilarly to Embodiment 1.

It is to be understood that the embodiments disclosed here isillustrative in all respects and not restrictive. The scope of thepresent invention is defined by the appended claims, and all changesthat fall within the meanings and the bounds of the claims, orequivalence of such meanings and bounds are intended to be embraced bythe claims.

It is to be noted that, as used herein and in the appended claims, thesingular forms “a”, “an”, and “the” include plural referents unless thecontext clearly dictates otherwise.

It is to be noted that the disclosed embodiment is illustrative and notrestrictive in all aspects. The scope of the present invention isdefined by the appended claims rather than by the description precedingthem, and all changes that fall within metes and bounds of the claims,or equivalence of such metes and bounds thereof are therefore intendedto be embraced by the claims.

1-8. (canceled).
 9. A non-transitory computer readable recording mediumstoring a computer program for causing a computer to execute processingcomprising: creating a table by a magnetic field analysis based on ananalytic model of an electromagnetic component including a coil, thetable including a flux linkage of the coil, an inductance of the coiland a current in the coil in association with one another, andsimulating behavior of the electromagnetic component at each of aplurality of time points by referring to the table using currents in thecoil calculated at a previous simulation step and at a step before theprevious simulation step.
 10. The non-transitory computer readablerecording medium according to claim 9, wherein the electromagneticcomponent is a motor including a plurality of coils, a stator and amover, the table is a look-up table storing flux linkages in theplurality of coils, inductances of the plurality of coils, currents inthe plurality of coils and a position of the mover in association withone another, and the program causes the computer to execute theprocessing of simulating the behavior of the motor by referring to thelook-up table using currents in the coils calculated at a previoussimulation step and at a step before the previous simulation step and aposition of the mover calculated at a previous simulation step.
 11. Thenon-transitory computer readable recording medium according to claim 10,wherein the program causes the computer to execute processing ofacquiring voltages to be applied to the coils of the motor from anexternal driving circuit simulator, reading out inductances of theplurality of coils and flux linkages of the coils by referring to thelook-up table using currents in the coils calculated at a previoussimulation step and a position of the mover at a present simulationstep, reading out inductances of the plurality of coils and fluxlinkages of the coils by referring to the look-up table using currentsin the coils calculated at a step before the previous simulation stepand a position of the mover at a previous simulation step, andcalculating currents of the coils at a present simulation step based onthe inductances of the coils and the flux linkages that are read out andthe voltages acquired.
 12. The non-transitory computer readablerecording medium according to claim 10, wherein the electromagneticcomponent is a star-connected motor, and the program causes the computerto execute processing of acquiring voltages to be applied to the coilsof the motor from an external driving circuit simulator, reading outinductances of the plurality of coils and flux linkages in the coils byreferring to the look-up table using currents in the coils calculated ata previous simulation step and a position of the mover at a presentsimulation step, reading out inductances of the plurality of coils andflux linkages of the coils by referring to the look-up table usingcurrents in the coils calculated at a step before the previoussimulation step and a position of the mover at a previous simulationstep, and calculating a voltage at a neutral point at a presentsimulation step based on the inductances of the coils and the fluxlinkages that are read out and the voltages acquired.
 13. Thenon-transitory computer readable recording medium according to claim 9,wherein the electromagnetic component is a motor including a pluralityof coils, and includes a stator and a mover, the plurality of coilsincludes U phase coil, V phase coil and W phase coil, the table is alook-up table storing flux linkages of the plurality of coils,inductances of the plurality of coils, currents in the plurality ofcoils and a position of the mover in association with one another, andthe program causes the computer to execute processing of acquiringvoltages to be applied to the coils of the motor from an externaldriving circuit simulator, reading out inductances of the plurality ofcoils and flux linkages of the coils by referring to the look-up tableusing currents in the coils calculated at a previous simulation step anda position of the mover at a present simulation step, reading outinductances of the plurality of coils and flux linkages of the coils byreferring to the look-up table using currents in the coils calculated ata step before the previous simulation step and a position of the moverat a previous simulation step, and calculating currents of the coils ata present simulation step based on Equation (1) below. $\begin{matrix}{{{\begin{pmatrix}{{R\;\Delta\; t} + {L_{uu}\left( {\text{?},\theta^{(n)}} \right)}} & {L_{uv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{uw}\left( {\text{?},\theta^{(n)}} \right)} \\{L_{vu}\left( {\text{?},\theta^{(n)}} \right)} & {{R\;\Delta\; t} + {L_{vv}\left( {\text{?},\theta^{(n)}} \right)}} & {L_{vw}\left( {\text{?},\theta^{(n)}} \right)} \\{L_{wu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{wv}\left( {\text{?},\theta^{(n)}} \right)} & {{R\;\Delta\; t} + {L_{ww}\left( {\text{?},\theta^{(n)}} \right)}}\end{pmatrix}\begin{pmatrix}I_{u}^{(n)} \\I_{v}^{(n)} \\I_{w}^{(n)}\end{pmatrix}} = {\begin{pmatrix}{\left( {V_{u\; 0}^{(n)} - V_{v\; 0}^{(n)}} \right)\Delta\; t} \\{\left( {V_{v\; 0}^{(n)} - V_{w\; 0}^{(n)}} \right)\Delta\; t} \\{\left( {V_{w\; 0}^{(n)} - V_{u\; 0}^{(n)}} \right)\Delta\; t}\end{pmatrix} + {\begin{pmatrix}{L_{uu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{uv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{uw}\left( {\text{?},\theta^{(n)}} \right)} \\{L_{vu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{vv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{vw}\left( {\text{?},\theta^{(n)}} \right)} \\{L_{wu}\left( {\text{?},\theta^{(n)}} \right)} & {L_{wv}\left( {\text{?},\theta^{(n)}} \right)} & {L_{ww}\left( {\text{?},\theta^{(n)}} \right)}\end{pmatrix}\begin{pmatrix}I_{u}^{({n - 1})} \\I_{v}^{({n - 1})} \\I_{w}^{({n - 1})}\end{pmatrix}} + {\begin{pmatrix}{L_{uu}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{uv}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{uw}\left( {\text{?},\theta^{({n - 1})}} \right)} \\{L_{vu}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{vv}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{vw}\left( {\text{?},\theta^{({n - 1})}} \right)} \\{L_{wu}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{wv}\left( {\text{?},\theta^{({n - 1})}} \right)} & {L_{ww}\left( {\text{?},\theta^{({n - 1})}} \right)}\end{pmatrix}\begin{pmatrix}{I_{u}^{({n - 1})} - I_{u}^{({n - 2})}} \\{I_{v}^{({n - 1})} - I_{v}^{({n - 2})}} \\{I_{w}^{({n - 1})} - I_{w}^{({n - 2})}}\end{pmatrix}} - \begin{pmatrix}{{\Psi_{u}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{u}\left( {\text{?},\theta^{({n - 1})}} \right)}} \\{{\Psi_{v}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{v}\left( {\text{?},\theta^{({n - 1})}} \right)}} \\{{\Psi_{w}\left( {\text{?},\theta^{(n)}} \right)} - {\Psi_{w}\left( {\text{?},\theta^{({n - 1})}} \right)}}\end{pmatrix}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (1)\end{matrix}$ where the upper-right superscript (n) means a value at then-th simulation step. R: electrical resistance value of each coil Δt:time between simulation steps L_(uu)(I

^((n)), θ^((n))): self-inductance of the U phase coil L_(uv)(I

^((n)), θ^((n))): mutual inductance between the U phase coil and the Vphase con L_(uw)(I

^((n)), θ^((n))): mutual inductance between the U phase coil and the Wphase coil L_(vv)(I

^((n)), θ^((n))): self-inductance of the V phase coil L_(vu)(I

^((n)), θ^((n))): mutual inductance between the V phase coil and the Uphase coil L_(vw)(I

^((n)), θ^((n))): mutual inductance between the V phase coil and the Wphase coil L_(ww)(I

^((n)), θ^(n))): self-inductance of the W phase coil L_(wu)(I

^((n)), θ^((n))): mutual inductance between the W phase coil and the Uphase coil L_(wv)(I

^((n)), θ^((n))): mutual inductance between the W phase coil and the Vphase coil V_(u0) ^((n)): voltage to be applied to the U phase coilV_(v0) ^((n)): voltage to be applied to the V phase coil V_(w0) ^((n)):voltage to be applied to of the W phase coil I

^((n)): (I_(u) ^((n)), I_(v) ^((n)), I_(w) ^((n))) I_(u) ^((n)) I_(v)^((n)), I_(w) ^((n)): current flowing in each coil θ^((n)): mechanicalangle of the mover Ψ_(u)(I

^((n)), θ^((n))): flux linkage in the U phase coil Ψ_(v)(I

^((n)), θ^((n))): flux linkage in the V phase coil Ψ_(w)(I

^((n)), θ^((n))): flux linkage in the W phase coil
 14. Thenon-transitory computer readable recording medium according to claim 9,wherein the electromagnetic component is a transformer, and the table isa look-up table storing a flux linkage of the coil, an inductance of thecoil and a current in the coil in association with one another, and theprogram causes the computer to execute processing of reading out aninductance of the coil and a flux linkage of the coil by referring tothe look-up table using a current in the coil calculated at a previoussimulation step, reading out an inductance of the coil and a fluxlinkage of the coil by referring to the look-up table using a current inthe coil calculated at a step before the previous simulation step, andcalculating a current of the coil at a present simulation step based onthe inductances and the flux linkages of the coil that are read out. 15.The non-transitory computer readable recording medium according to claim14, wherein the program causes the computer to execute processing ofcalculating currents of the coils at a simulation step based on Equation(2) below. $\begin{matrix}{{{\begin{pmatrix}{{L\left( \text{?} \right)} + {R\;\Delta\; t}} & {1\Delta\; t} \\1^{T} & 0\end{pmatrix}\begin{pmatrix}I^{(n)} \\0\end{pmatrix}} = {\begin{pmatrix}{V_{0}^{(n)}\Delta\; t} \\0\end{pmatrix} + {\begin{pmatrix}{L\left( \text{?} \right)} & 0 \\0 & 0\end{pmatrix}\begin{pmatrix}I^{({n - 1})} \\0\end{pmatrix}} + {\begin{pmatrix}{L\left( \text{?} \right)} & 0 \\0 & 0\end{pmatrix}\begin{pmatrix}{I^{({n - 1})} - I^{({n - 2})}} \\0\end{pmatrix}} - \begin{pmatrix}{{\Psi\left( \text{?} \right)} - {\Psi\left( \text{?} \right)}} \\0\end{pmatrix}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (2)\end{matrix}$ where the upper-right superscript (n) means a value at then-th simulation step. (L(?^((n))))_(ij) = L_(ij)(?^((n)))(R)_(ij) = R_(i)δ_(ij) 1 = (1, 1 . . . 1)^(T) : x − order  unit  vectorI^((n)) = (I₁^((n)), I₂^((n)), . . . , I_(x)^((n)))^(T)V₀^((n)) = (V₁₀^((n)), V₂₀^((n)), . . . , V_(x0)^((n)))^(T)Ψ(?^((n))) = (Ψ₁(?^((n))), Ψ₂(?^((n))), . . . , Ψ_(x)(?^((n))))^(T)?indicates text missing or illegible when filed ij: 1,2, . . . , χ χ:the number of phases R_(t): electrical resistance of each coil Δt: timebetween simulation steps δ_(ij): Kronecker's delta L_(ij)(I

^((n))): self-inductance (i=j) of the i phase coil or mutual inductance(i≠j) between the i phase coil and the j phase coil I

^((n)): (I₁ ^((n)), I₂ ^((n)), . . . , I₁₀₂ ^((n))) I₁ ^((n)), I₂^((n)), . . . , I₁₀₂ ^((n)): current flowing in each coil V₁₀ ^((n)),V₂₀ ^((n)), . . . , V_(χ0) ^((n)): voltage to be applied to each coilΨ₁, Ψ₂, . . . , Ψ_(χ):flux linkage in each coil
 16. The non-transitorycomputer readable recording medium according to claim 9, wherein theprogram causes the computer to execute processing of simulating thebehavior of the electromagnetic component at a present simulation stepby solving a voltage equation related to the coil without performingrepetitive calculation.
 17. A simulation method causing a computer toexecute processing comprising: creating a table by a magnetic fieldanalysis based on an analytic model of an electromagnetic componentincluding a coil, the table including a flux linkage of the coil, aninductance of the coil and a current in the coil in association with oneanother; and simulating behavior of the electromagnetic component ateach of a plurality of time points by referring to the table usingcurrents in the coil calculated at a previous simulation step and at astep before the previous simulation step.
 18. A simulation devicecomprising an arithmetic unit, the arithmetic unit creating a table by amagnetic field analysis based on an analytic model of an electromagneticcomponent including a coil, the table including a flux linkage of thecoil, an inductance of the coil and a current in the coil in associationwith one another, and simulating behavior of the electromagneticcomponent at each of a plurality of time points by referring to thetable using currents in the coil calculated at a previous simulationstep and at a step before the previous simulation step.